An Effective Diffusion Model

Effective diffusion models have also been used to account for intermediate degrees of mixing in the axial direction — see Pavlica and Olson [1970] for a comprehensive survey. An example of such a model is developed here for the case of a reaction catalyzed by a solid and no reaction in the liquid. 

Steady-state continuity equation for A in the liquid phase  

The fourth term represents the transfer of A from the liquid to the catalyst surface, where is the liquid-solid interfacial area per unit reactor volume (in 

Transfer from liquid to catalyst surface and reaction  

When internal catalyst pellet concentration gradients have to be accounted for, the right-hand side of (14.2.4-3) would have to be multiplied by rj, the effectiveness factor, computed as described in Chapters 3 and 11. Accounting for temperature gradients in the axial direction would require an additional differential heat balance, analogous in structure to (14.2.4-2). 

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Suppose that catalyst pellets in the shape of right-circular cylinders have a measured effectiveness factor of r] when used in a packed-bed reactor for a first-order reaction. In an effort to increase catalyst activity, it is proposed to use a pellet with a central hole of radius i /, < Rp. Determine the best value for RhjRp based on an effective diffusivity model similar to Equation (10.33). Assume isothermal operation ignore any diffusion limitations in the central hole, and assume that the ends of the cylinder are sealed to diffusion. You may assume that k, Rp, and eff are known. 

SOLUTION OF MULTICOMPONENT DIFFUSION PROBLEMS USING AN EFFECTIVE DIFFUSIVITY MODEL... 

Example 6.3.1 Computation ofthe Fluxes with an Effective Diffusivity Model... 

Here we use an effective diffusivity model for the diffusion fluxes... 

Diffusional interaction effects are quite important in this example. We leave it as an exercise for our readers to determine the molar rates of condensation using an effective diffusivity model. It is worth pointing out, however, that the rates are quite different from those calculated here. 

The theory of seaweed formation does not only apply to solidification processes but in fact to the completely different phenomenon of a wettingdewetting transition. To be precise, this applies to the so-called partial wetting scenario, where a thin liquid film may coexist with a dry surface on the same substrate. These equations are equivalent to the one-sided model of diffusional growth with an effective diffusion coefficient which depends on the viscosity and on the thermodynamical properties of the thin film. 

The term numerical diffusion describes the effect of artificial diffusive fluxes which are induced by discretization errors. This effect becomes visible when the transport of quantities with small diffusivities [with the exact meaning of small yet to be specified in Eq. (42)] is considered. In macroscopic systems such small diffusivities are rarely found, at least when being looked at from a phenomenological point of view. The reason for the reduced importance of numerical diffusion in many macroscopic systems lies in the turbulent nature of most macro flows. The turbulent velocity fluctuations induce an effective diffusivity of comparatively large magnitude which includes transport effects due to turbulent eddies [1]. The effective diffusivity often dominates the numerical diffusivity. In contrast, micro flows are often laminar, and especially for liquid flows numerical diffusion can become the major effect limiting the accuracy of the model predictions. 

One must understand the physical mechanisms by which mass transfer takes place in catalyst pores to comprehend the development of mathematical models that can be used in engineering design calculations to estimate what fraction of the catalyst surface is effective in promoting reaction. There are several factors that complicate efforts to analyze mass transfer within such systems. They include the facts that (1) the pore geometry is extremely complex, and not subject to realistic modeling in terms of a small number of parameters, and that (2) different molecular phenomena are responsible for the mass transfer. Consequently, it is often useful to characterize the mass transfer process in terms of an effective diffusivity, i.e., a transport coefficient that pertains to a porous material in which the calculations are based on total area (void plus solid) normal to the direction of transport. For example, in a spherical catalyst pellet, the appropriate area to use in characterizing diffusion in the radial direction is 47ir2. 

The measured value of k Sg is 0.716 cm3/(sec-g catalyst) and the ratio of this value to k ltTueSg should be equal to our assumed value for the effectiveness factor, if our assumption was correct. The actual ratio is 0.175, which is at variance with the assumed value. Hence we pick a new value of rj and repeat the procedure until agreement is obtained. This iterative approach produces an effectiveness factor of 0.238, which corresponds to a differs from the experimental value (0.17) and that calculated by the cylindrical pore model (0.61). In the above calculations, an experimental value of eff was not available and this circumstance is largely responsible for the discrepancy. If the combined diffusivity determined in Illustration 12.1 is converted to an effective diffusivity using equation 12.2.9, the value used above corresponds to a tortuosity factor of 2.6. If we had employed Q)c from Illustration 12.1 and a tortuosity factor of unity to calculate eff, we would have determined that rj = 0.65, which is consistent with the value obtained from the straight cylindrical pore model in Illustration 12.2. 

The terms Jga and Jsa are the diffusive fluxes of species a in the gas and solid phases, respectively. Note that in addition to molecular-scale diffusion, these terms include dispersion due to particle-scale turbulence. The latter is usually modeled by introducing a gradient-diffusion model with an effective diffusivity along the lines of Eqs. (149) and (151). Thus, for large particle Reynolds numbers the molecular-scale contribution will be negligible. The term Ma is the... 

The low-temperature enhanced diffusion of B can be modeled by calculating an effective diffusivity that is then applied to the calculation of the B profile by using the PREDICT program (59). The duration of enhanced diffusion is related to the damage annealing time. Empirically, the removal... 

Mass transport inside the catalyst has been usually described by applying the Fick equation, by means of an effective diffusivity Deff a Based on properties of the interface and neglecting the composition effect, composite diffusivity of the multi-component gas mixture is calculated through the simplified Wilke model [13], The effect of pore-radius distribution on Knudsen diffusivity is taken into account. The effective diffusivity DeffA is given by... 

The effective-scale models have been most often used in the description of transport and reaction processes within the porous structure of catalysts. Such models are based on the introduction of an effective diffusion coefficient De, that is used in the analogy to the Fick s law for the description of diffusion... 

The extraction of toluene and 1,2 dichlorobenzene from shallow packed beds of porous particles was studied both experimentally and theoretically at various operating conditions. Mathematical extraction models, based on the shrinking core concept, were developed for three different particle geometries. These models contain three adjustable parameters an effective diffusivity, a volumetric fluid-to-particle mass transfer coefficient, and an equilibrium solubility or partition coefficient. K as well as Kq were first determined from initial extraction rates. Then, by fitting experimental extraction data, values of the effective diffusivity were obtained. Model predictions compare well with experimental data and the respective value of the tortuosity factor around 2.5 is in excellent agreement with related literature data. 

The existing models can be grouped in two principal categories, black box models and structural models. Within the empirical black box models the membrane is considered a continuous, nonporous phase in which water of hydration is dissolved. An effective diffusion coefficient which is a characteristic function of the water content controls the water flux. 

In contrast to the diffusion approach, in the previous sections hydraulic permeation was considered as the effective mode of water transport. Transformed to the form of an effective diffusion coefficient the transport coefficient of the latter model becomes... 

The most rigorous formulation to describe adsorbate transport inside the adsorbent particle is the chemical potential driving force model. A special case of this model for an isothermal adsorption system is the Fickian diffusion (FD), model which is frequently used to estimate an effective diffusivity for adsorption of component i (D,) from experimental uptake data for pure gases.The FD model, however, is not generally used for process design because of mathematical complexity. A simpler analytical model called linear driving force (LDF) model is often used. ° According to this model, the rate of adsorption of component i of a gas mixture... 

This allows us to make a homogeneous model of the porous catalyst pellet in which we now have a diffusive flux given as the product of an effective diffusion coefficient and a concentration gradient, and a rate of reaction given by the product of the catalytic area and the reaction rate per unit area. 

If the nonzero fluxes have the same sign (i.e., they are all in the same direction), then effective diffusivity methods are more likely to give reasonable results. This is nearly always the case in condensation and absorption processes and this goes some way at least to explaining why effective diffusivity methods usually give good estimates of the total amount condensed and the total heat load even if the individual condensation rates are not so well predicted. Webb et al. (1981) discussed in detail the conditions that must apply for an effective diffusivity method to be a useful model in multicomponent condensation. 

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